Ok, this is just to get you started. I'm way too tired to continue, and i think i made a mistake somewhere in the calculations and i'm too lazy to go over it at the moment--time to sleep. anyway, here is what i have so far.

We see that a(n) = 2^n + n! holds for n = 0, 1, 2. We now show that a(n) = 2^n + n! for n > 2 by the Strong Form of Induction.